How do you integrate #int e^xe^x# using substitution?

Answer 1
#inte^xe^xdx#

Method 1 - Immediate Substitution

Make the substitution #u=e^x#, which implies that #du=e^xdx#, so
#inte^xe^xdx=int(e^x)(e^xdx)=intudu=u^2/2=(e^x)^2/2=e^(2x)/2+C#

Method 2 - Simplification, then Substitution

Use the rule #a^b(a^c)=a^(b+c)# to rewrite the integral as
#inte^xe^xdx=inte^(2x)dx#
Now substitute #u=2x# so #du=2dx#:
#inte^(2x)dx=1/2int(e^(2x))(2dx)=1/2inte^udu#
Since #inte^udu=e^u#:
#1/2inte^udu=1/2e^u=e^u/2=e^(2x)/2+C#
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Answer 2

To integrate ( \int e^x e^{x} ) using substitution, let ( u = e^x ). Then, ( du/dx = e^x ) or ( du = e^x dx ). Substituting ( u ) and ( du ), the integral becomes ( \int u , du ).

Now integrate ( u ) with respect to ( u ), which gives ( \frac{u^2}{2} ). Then, resubstitute ( e^x ) for ( u ), yielding ( \frac{(e^x)^2}{2} + C ), where ( C ) is the constant of integration.

So, ( \int e^x e^{x} , dx = \frac{(e^x)^2}{2} + C ).

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Answer 3

To integrate (\int e^x e^{2x} , dx) using substitution, let (u = e^x). Then, (du = e^x , dx).

Substitute these into the integral: [\int e^x e^{2x} , dx = \int u e^{2x} , du]

Now, the integral is in terms of (u), so we can integrate it with respect to (u): [= \int u e^{2x} , du] [= \int u e^{2x} , du] [= \int u , du] [= \frac{1}{2} u^2 + C]

Finally, substitute back for (u) to get the final result: [= \frac{1}{2} (e^x)^2 + C] [= \frac{1}{2} e^{2x} + C]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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